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Chapter 2: Problem 33

Students in an AP Statistics class were asked how many hours of television they watch per week (including online streaming). This sample yielded an average of 4.71 hours, with a standard deviation of 4.18 hours. Is the distribution of number of hours students watch television weekly symmetric? If not, what shape would you expect this distribution to have? Explain your reasoning.

Short Answer

Expert verified
The distribution is likely right-skewed, indicating some students watch significantly more TV than average.

Step by step solution

01

Review Given Data

We have an average (mean) number of hours of television watched as 4.71 hours per week, and a standard deviation of 4.18 hours. These are numerical summaries of our data, which help us analyze its distribution.
02

Consider Symmetrical Distribution Criteria

A distribution is symmetric if it is mirrored around its mean. In a perfectly normal distribution, mean = median = mode. However, at this step, we only have the mean and the standard deviation, so one way to hypothesize the symmetry is to check the relationship between the mean and the expected median.
03

Assess Skewness with Mean and Standard Deviation

The high standard deviation relative to the mean (4.18 compared to 4.71) suggests that there is considerable disparity in the data values. This disparity means that the data is spread out, but without more data, we cannot conclusively determine symmetry or exact skewness direction.
04

Hypothesize if Skewness Exists

Since mean > median in a right-skewed (positively skewed) distribution and mean < median in a left-skewed (negatively skewed) distribution, if the distribution is not symmetric, we might guess the type of skew based on the spread suggested by the high standard deviation. Given the values, it is more plausible that the distribution is right-skewed.
05

Justify Using Real-World Context

In real-life scenarios, it's common to see that a few students watch a lot more television than their peers causing the mean to increase, which suggests a right-skewed distribution. Therefore, given the mean is higher than what might be expected as a median, the distribution is likely right-skewed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distribution Symmetry
When we talk about distribution symmetry, we're referring to how data points in a dataset are arranged around the center. If a distribution is symmetric, it looks the same on both sides when you fold it at the mean. Think of a perfect bell shape, where data is evenly spread out around a central value. In that case, the mean, median, and mode are all the same or very close.
  • A symmetric distribution tells us that most data points fall uniformly around the central peak.
  • For the television hours watched, if the mean were equal to the median, it would suggest symmetry.
Without the median given, we check if the mean can be an indicator. The problem hints at assessing symmetry by using the mean and expected behaviors of data.
Standard Deviation
Standard deviation gives us an idea of the spread of the data. A smaller standard deviation means the data points tend to be closer to the mean, while a larger value implies more spread out data.
  • In the television watching example, the standard deviation is 4.18, which is close to the mean of 4.71 hours.
  • This relatively large spread indicates variability in how much television different students watch.
The larger the standard deviation compared to the mean, as in this example, the more likely the data spreads out significantly from the average, hinting at potential skewness or asymmetry in the data.
Right-Skewed Distribution
A right-skewed distribution, also called positively skewed, is when the data's tail on the right side is longer or fatter than on the left. This happens when there are a few exceptionally high values pulling the mean to the right.
  • A characteristic of right-skewed distributions is that the mean is usually greater than the median.
  • In our example, the average hours watched being 4.71 hours doesn鈥檛 directly qualify its position to the median, but it鈥檚 implied that the mean is elevated by outliers who watch a lot more TV, suggesting positive skewness.
In real-life contexts, like our exercise, this skew indicates that while most students watch a few hours, a small number maybe watch a lot more, tilting the mean upwards.
Mean versus Median
Comparing mean and median helps us understand the data's distribution more deeply. The mean is a simple average, but it can be distorted by extreme values, unlike the median, which is the middle value when data is sorted.
  • In a perfect symmetric distribution, mean and median are equal.
  • However, a right-skewed distribution usually has a mean greater than the median.
In the exercise about television hours, the hint is that if the distribution is not symmetric, it might be skewed right because extreme high values could make the mean higher than a potential median. This difference tells us more about the shape and skewness of the data.

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Most popular questions from this chapter

Below are the final exam scores of twenty introductory statistics students. $$57,66,69,71,72,73,74,77,78,78,79,79,81,81,82,83,83,88,89,94$$ Create a box plot of the distribution of these scores. The five number summary provided below may be useful. $$ \begin{array}{ccccc} \text { Min } & \text { Q1 } & \text { Q2 (Median) } & \text { Q3 } & \text { Max } \\ \hline 57 & 72.5 & 78.5 & 82.5 & 94 \end{array} $$

A baseball team's success in a season is usually measured by their number of wins. In order to win, the team has to have scored more points (runs) than their opponent in any given game. As such, number of runs is often a good proxy for the success of the team. The table below shows number of runs, home runs, and batting averages for a random sample of 10 teams in the 2014 Major League Baseball season. \(^{16}\) $$\begin{array}{rlrrr} \hline & \text { Team } & \text { Runs } & \text { Home runs } & \text { Batting avg. } \\ \hline 1 & \text { Baltimore } & 705 & 211 & 0.256 \\ 2 & \text { Boston } & 634 & 123 & 0.244 \\ 3 & \text { Cincinnati } & 595 & 131 & 0.238 \\ 4 & \text { Cleveland } & 669 & 142 & 0.253 \\ 5 & \text { Detroit } & 757 & 155 & 0.277 \\ 6 & \text { Houston } & 629 & 163 & 0.242 \\ 7 & \text { Minnesota } & 715 & 128 & 0.254 \\ 8 & \text { NY Yankees } & 633 & 147 & 0.245 \\ 9 & \text { Pittsburgh } & 682 & 156 & 0.259 \\ 10 & \text { San Francisco } & 665 & 132 & 0.255 \\ \hline \end{array}$$ (a) Draw a scatterplot of runs vs. home runs. (b) Draw a scatterplot of runs vs. batting averages. (c) Are home runs or batting averages more strongly associated with number of runs? Explain your reasoning.

Facebook data indicate that \(50 \%\) of Facebook users have 100 or more friends, and that the average friend count of users is \(190 .\) What do these findings suggest about the shape of the distribution of number of friends of Facebook users? \(^{40}\)

For each of the following, state whether you expect the distribution to be symmetric, right skewed, or left skewed. Also specify whether the mean or median would best represent a typical observation in the data, and whether the variability of observations would be best represented using the standard deviation or IQR. Explain your reasoning. (a) Housing prices in a country where \(25 \%\) of the houses cost below $$\$ 350,000,50 \%$$ of the houses cost below $$\$ 450,000,75 \%$$ of the houses cost below $$\$ 1,000,000$$ and there are a meaningful number of houses that cost more than $$\$ 6,000,000$$ (b) Housing prices in a country where \(25 \%\) of the houses cost below $$\$ 300,000,50 \%$$ of the houses cost below $$\$ 600,000,75 \%$$ of the houses cost below $$\$ 900,000$$ and very few houses that cost more than $$\$ 1,200,000 .$$ (c) Number of alcoholic drinks consumed by college students in a given week. Assume that most of these students don't drink since they are under 21 years old, and only a few drink excessively. (d) Annual salaries of the employees at a Fortune 500 company where only a few high level executives earn much higher salaries than all the other employees.

Below are the final exam scores of twenty introductory statistics students. $$79,83,57,82,94,83,72,74,73,71,66,89,78,81,78,81,88,69,77,79$$ Draw a histogram of these data and describe the distribution.
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